When to use
When specifying a thermowell for a new line, when checking a vendor's wake frequency calculation, when a process change raises the flow (debottlenecking, start-up, steam blow) and when a well has already failed and the immersion length must be revised. It covers straight, tapered and step-shank wells, in flanged, lap-joint, threaded, socket-weld and weld-in mounting.
Why thermowells fail
A thermowell is a cantilever in cross-flow. Vortices shed alternately from each side of the tip at the Strouhal frequency fs and push the well sideways; the drag force oscillates at twice that frequency and pushes it in line with the flow. If either excitation coincides with the natural frequency of the installed well, the amplitude grows until the root cracks by high-cycle fatigue. The failure is fast and the broken tip travels downstream into pumps, control valves and turbines — which is why the check is mandatory, not optional.
The four criteria of TW-2016
The 2016 edition replaced the single frequency ratio of the 1974 method with four independent checks, and one failure is enough to reject the well:
- Frequency — r = fs/fnc below 0.4, or below 0.8 outside the 0.4–0.6 band when the in-line resonance stress passes.
- Dynamic stress — the oscillating bending stress at the root, magnified near resonance and multiplied by Kt, against the temperature-corrected fatigue limit.
- Static stress — steady drag plus pressure stresses, combined by von Mises, against 1.5 times the allowable stress.
- External pressure — the smallest of the shank, tip and flange ratings.
The tool also checks the dimensional scope of Tables 4-1-1 and 4-2-1: outside it the correlations are not valid and no verdict is issued.
The natural frequency, step by step
fa is the textbook cantilever frequency with the average diameter. TW-2016 then corrects it: Hf for the tapered or stepped profile, Ha,f for the fluid that moves with the well, Ha,s for the sensor inside it (the thermocouple or RTD), and Hc for the fact that the root is not perfectly rigid. In the example, these factors take 133.9 Hz to 163.3 Hz. For step-shank wells only the two tip diameters with published coefficients — 22.23 mm and 12.70 mm — can be calculated; the standard forbids interpolation.
Velocity and pressure ratings
Instead of only “pass” or “fail”, the tool reports the §9-2 rating pair: the highest velocity at which the well passes everything from zero upward, and the pressure rating, along with the governing criterion. Compare the velocity rating with the line velocity at the maximum expected flow (§6-3.3), upsets included — a relief valve lifting, for example — computed from flow and bore just as in magnetic flow meter sizing. With a corrosion allowance it evaluates the three geometries of §6-2 and reports the minimum. A vendor-rating mode applies Ha,f = 1 and the default sensor density, as §9-2 requires for ratings published without known fluid properties.
The tool applies three readings of the printed equations that differ from a literal transcription — the exponent in eq. 6-5-2, Ls² in eq. 6-10-10 and the position of the 0.0833 term in eq. 6-13-1 — each confirmed by reproducing the Section 8 worked examples.
What this tool does not cover
E(T), S(T) and ρm are entered by the user from the governing code (B31.1, B31.3); they are not built into the tool. Support collars and elbow installations with the tip pointing upstream are outside the scope of the standard and get no verdict. The standard itself excludes wells fabricated from pipe, welded overlays, closely spaced wells, broadband turbulence, pulsating flow, pipe-borne vibration and the thermal response of the sensor. Supercritical operation (r ≥ 1) and pressures above 103 MPa are flagged, not calculated.
Formulas and fundamentals
Re = ρ·V·B/μ ; Ns = 0.22(1 − 22/Re) for 22 ≤ Re < 1300 ; Ns = 0.213 − 0.0248x² + 0.0095x³, x = log10(Re/1300), for Re < 5×10⁵ ; Ns = 0.22 above Ns is evaluated at the tip diameter B. With viscosity unknown, or with the correlation switched off, the tool uses the conservative constant 0.22.
fs = Ns · V / B Transverse (lift) excitation at fs; the in-line (drag) excitation acts at 2·fs, which is why in-line resonance appears at roughly half the transverse resonance velocity.
fa = (1.875² / 2π) · √(E·I / m) / L² ; I = π(Da⁴ − d⁴)/64 ; m = ρm·π(Da² − d²)/4 Cantilever first mode with the average diameter Da (A for straight, (A+B)/2 for tapered, A for step-shank). E and ρm are the metal properties at operating temperature. The L² in the denominator is why length dominates.
fnc = Hc · Hf · Ha,f · Ha,s · fa ; Ha,f = 1 − ρ/(2ρm) ; Ha,s = 1 − (ρs/2ρm)/[(Da/d)² − 1] Hf corrects the profile (eq. 6-5-2 for straight/tapered, eq. 6-5-3 with the Table 6-5.3-1 coefficients for 22.23 mm or 12.70 mm step-shank tips), Ha,f and Ha,s add the mass of fluid and sensor (ρs = 2700 kg/m³ by default) and Hc the foundation compliance — Hc = 1 − 0.61(A/L)/(1 + 1.5b/A)² with a root fillet b, or 1 − 0.9A/L for threaded wells.
Nsc = π²·ζ·(ρm/ρ)·[1 − (d/B)²] ; limit r < 0.8 if Nsc > 2.5 and Re < 10⁵ ; otherwise r < 0.8 outside 0.4–0.6 if the in-line stress passes, else r < 0.4 ζ = 0.0005 (intrinsic damping, conservative). High mass-damping in a low density gas suppresses in-line resonance; in liquids Nsc is tiny and the in-line check decides the limit.
V_IR ≈ B·fnc/(2·Ns) ; So,max,IR = Kt · G · [1/(2ζ)] · ½ρ·Cd·V_IR² ≤ FT·FE·Sf With ζ = 0.0005 the magnification is 1000. Cd = 0.1. If this stress exceeds the fatigue limit, the well must stay below 0.4 fnc.
S_D = G · ½ρ·C_D·V² (C_D = 1.4) ; Smax = S_D + Sa ; √{[(Smax−Sr)² + (Smax−St)² + (St−Sr)²]/2} ≤ 1.5·S G is the geometric bending parameter at the support plane; for a tapered well with no shielded length G = 16L²(1 + 2B/A) / [3πA²(1 − (d/A)⁴)]. Sr, St and Sa are the radial, tangential and axial pressure stresses; S is the allowable stress from the governing code.
So,max = Kt · √(Sd² + Sl²) ; Sd = G·FM'·½ρ(0.1)V² ; Sl = G·FM·½ρ(1.0)V² ; FM = 1/(1 − r²) ; FM' = 1/(1 − 4r²) ≤ FT·FE·Sf Kt = 1.1 + 0.033·(A/b) up to 2.2 (2.3 for threaded roots). Sf is the fatigue endurance limit at 10¹¹ cycles from Table 6-12.3-1 (class A or B material, as-welded, welded and machined, or no weld), FT = E/Eref the temperature correction and FE the environmental factor.
Pc = 0.66·S·[2.167/(2B/(B − d)) − 0.0833] ; Pt = (S/0.13)·(t/d)² ; Pr = min(Pc, Pt, Pf) ≥ P Shank and tip ratings; for flanged and lap-joint wells the flange rating Pf enters the minimum. Above 103 MPa the standard sends the designer elsewhere and the tool flags it.
Standards & methods
- ASME PTC 19.3 TW-2016 (reaffirmed 2025) — Thermowells
- ASME B31.1 / B31.3 — source of E, S and ρm at temperature (entered by the user)
- ASME B16.5 — flange pressure rating Pf
- ASME BPVC Section VIII Div. 1, UG-28 — optional shank external pressure rating
Typical reference values
| Quantity | Typical range | Note |
|---|---|---|
| Frequency ratio r = fs/fnc, liquids | below 0.4 | in-line resonance stress usually fails with ζ = 0.0005 |
| Frequency ratio, in-line stress passes | below 0.8, never steady inside 0.4–0.6 | — |
| Strouhal number Ns | 0.22 (Re ≥ 5×10⁵) · ≈ 0.19–0.20 for Re 10⁴–10⁵ | — |
| Intrinsic damping ζ (default) | 0.0005 | — |
| Low-velocity screening (§6-3.6) | V < 0.64 m/s, L ≤ 0.61 m, A − d ≥ 9.55 mm, B ≥ 12.7 mm, S ≥ 69 MPa, Sf ≥ 21 MPa | frequency and dynamic checks waived; static and pressure kept |
| Fatigue limit Sf, class B (austenitic) | 37.2 / 62.8 / 93.8 MPa | as-welded / welded and machined / no weld |
| Dimensional scope, straight and tapered | L 63.5–609.6 mm · d 3.18–20.96 mm · B 9.2–46.5 mm · B/A 0.58–1 · d/B 0.16–0.71 · wall and tip ≥ 3 mm | — |
Worked example
Tapered flanged thermowell in a water line
Inputs
- Fluid — water at 50 °C, ρ / μ
- 990 / 0.55 kg/m³ / cP
- Maximum velocity / pressure
- 4 / 10 m/s / bar
- Profile and mounting
- tapered, flanged —
- Root A / tip B / bore d / tip t
- 27 / 19 / 6.6 / 6.4 mm
- Unsupported length L / root fillet b
- 350 / 3 mm
- Material — 316 SS, class B, welded and machined; E / S / ρm
- 192 000 / 138 / 8000 MPa / MPa / kg/m³
- Flange rating Pf (entered)
- 40 bar
Results
- Reynolds / Strouhal
- 136 800 / 0.190 —
- Installed natural frequency fnc (fa 133.9 Hz)
- 163.3 Hz
- Wake frequency fs
- 40.0 Hz
- Frequency ratio r = fs/fnc (limit 0.4)
- 0.245 —
- Stress at in-line resonance vs fatigue limit
- 3006 vs 61.8 MPa (fails → limit 0.4 fnc)
- Dynamic stress So,max vs fatigue limit
- 8.18 vs 61.8 MPa
- Static von Mises vs 1.5·S
- 7.64 vs 207 MPa
- Pressure rating (flange governs; shank 568 bar)
- 40 bar
- Velocity rating (governed by frequency)
- 6.37 m/s
The well passes all four criteria at 4 m/s. The stresses have a wide margin; what limits it is frequency. With a Scruton number of 0.035, the in-line resonance stress would be about 3000 MPa against a 61.8 MPa fatigue limit, so the 0.4 fnc limit applies, and the velocity rating is 6.37 m/s. At 6.5 m/s r becomes 0.409 and the well fails. Length is the lever: with the same section the rating is 4.97 m/s at L = 400 mm, 8.42 m/s at 300 mm and 11.6 m/s at 250 mm.
Common mistakes
- Using the normal flow velocity. §6-3.3 asks for the maximum expected velocity, including start-up, upset, relief and steam blow. A well rated at 6.4 m/s is not safe in a line that sees 7 m/s during a blowdown.
- Applying the 0.8 limit of the 1974 edition in a liquid. With a Scruton number near 0.04 the in-line resonance stress fails by two orders of magnitude, and the TW-2016 limit drops to 0.4 fnc.
- Taking E and S at room temperature. Both fall with temperature; a lower E lowers fnc, and the fatigue limit is scaled by FT = E/Eref.
- Treating a threaded well like a flanged one. Thread roots have Kt ≥ 2.3, zero fillet radius and their own compliance Hc = 1 − 0.9A/L; §6-1.2.1 lists avoiding threaded mounting as a way to improve strength.
- Ignoring the corrosion allowance. §6-2 requires three geometries — nominal, root and t reduced, tip and t reduced — and the rating is the minimum.
- Adding a support collar to "fix" a long well. Support collars are outside the scope of the standard (§6-7(e)); the tool returns no verdict for them.
Frequently asked questions
What is a thermowell wake frequency calculation?
It compares the frequency at which the flow sheds vortices behind the well (the wake or Strouhal frequency, fs = Ns·V/B) with the natural frequency of the installed well. ASME PTC 19.3 TW-2016 adds stress and pressure checks, so the full calculation has four criteria, and any one of them can reject the well.
Why 0.4 and not 0.8?
The 0.8 limit is only allowed if the well survives the in-line (drag) resonance, which happens near r = 0.5. In liquids the mass-damping is so low that the resonance stress is far above the fatigue limit, and the well must stay below 0.4 fnc. Low-density gases with Nsc above 2.5 and Re below 10⁵ can use 0.8 directly.
What is the velocity rating?
The highest velocity up to which the four criteria pass over the whole range from zero — the plant can ramp up to it without crossing a forbidden band. The tool reports it together with the pressure rating and the governing criterion, as §9-2 asks, because that number is useful even when the well fails.
How do I raise the velocity rating?
Shorten the well first: fnc falls roughly with L² — in the example, going from 400 to 350 mm raises fnc by 30 %. Then thicken the root (tapered or step-shank profile), avoid threaded mounting and add a root fillet to lower Kt.
When can the frequency check be skipped?
Under the §6-3.6 low-velocity screening: velocity below 0.64 m/s, a robust well no longer than 0.61 m, and material with S ≥ 69 MPa and Sf ≥ 21 MPa, without stress corrosion. Static stress and external pressure are still required.
Has the tool been checked against the standard?
Yes. It reproduces, value by value, both worked examples of Section 8 — the tapered weld-in well in steam (US Customary) and the threaded step-shank well in hot water (SI).
Glossary
- Wake frequency (fs)
- Vortex shedding frequency behind the well, fs = Ns·V/B; the transverse excitation.
- Strouhal number (Ns)
- Dimensionless shedding frequency, a function of Reynolds; 0.22 at high Re.
- Installed natural frequency (fnc)
- First bending mode of the well corrected for profile, fluid and sensor mass and mounting compliance.
- In-line resonance
- Resonance driven by the drag force, which oscillates at 2·fs; it occurs near r = 0.5.
- Scruton number (Nsc)
- Mass-damping parameter π²ζ(ρm/ρ)[1 − (d/B)²]; high values suppress in-line resonance.
- Unsupported length (L)
- Length from the support plane to the tip, exposed to bending.
- Shielded length (L0)
- Part of L inside the nozzle, out of the flow; it reduces the bending moment.
- Velocity rating
- Highest velocity at which the well meets all four criteria from zero up to it.